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<p>We present a generalization of Cantor’s ternary set and ternary function through the Cantor expansion of real numbers. Our analysis demonstrates that the Hausdorff dimension of these generalized sets and the Hölder regularity of the corresponding functions are influenced by the specific sequence used to define the expansion. Additionally, we explore the measure-theoretic properties of these sets through their Hausdorff <i>h</i>-measure, highlighting the connections between numeral systems and fractal geometry.</p>

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On the fractal properties of generalized Cantor sets and Devil’s staircase functions

  • 期刊论文

  • 发表日期：

    2025年6月16日

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  • Thomas Devos,
  • Laurent Loosveldt,
  • Samuel Nicolay

摘要

We present a generalization of Cantor’s ternary set and ternary function through the Cantor expansion of real numbers. Our analysis demonstrates that the Hausdorff dimension of these generalized sets and the Hölder regularity of the corresponding functions are influenced by the specific sequence used to define the expansion. Additionally, we explore the measure-theoretic properties of these sets through their Hausdorff h-measure, highlighting the connections between numeral systems and fractal geometry.